How to work with Git branches and Rails migrations

I am working on a rails app with quite a few git branches and many of them include db migrations. We try to be careful but occasionally some piece of code in master asks for a column that got removed/renamed in another branch. What would be a nice solution to "couple" git branches with DB states? What would these "states" actually be? We can't just duplicate a database if it's a few GBs in size. A